Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 545120, 9 pages
doi:10.1155/2012/545120
Review Article
Numerical Simulation of Unsteady
Compressible Flow in Convergent Channel:
Pressure Spectral Analysis
Petra Pořı́zková,1 Karel Kozel,2 and Jaromı́r Horáček2
1
Department of Technical Mathematics, Faculty of Mechanical Engineering,
Czech Technical University in Prague, Karlovo Nám. 13, 121 35 Praha 2, Czech Republic
2
Institute of Thermomechanics AS CR, Dolejškova 5, 18200 Prague 8, Czech Republic
Correspondence should be addressed to Petra Pořı́zková, puncocha@marian.fsik.cvut.cz
Received 16 January 2012; Accepted 8 March 2012
Academic Editor: Fu-Yun Zhao
Copyright q 2012 Petra Pořı́zková et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
This study deals with the numerical solution of a 2D unsteady flow of a compressible viscous
fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a
prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing
the channel during oscillations. The flow is described by the system of Navier-Stokes equations
for laminar flows. The numerical solution is implemented using the finite volume method FVM
and the predictor-corrector Mac-Cormack scheme with Jameson artificial viscosity using a grid
of quadrilateral cells. Due to the motion of the grid, the basic system of conservation laws is
considered in the arbitrary Lagrangian-Eulerian ALE form. The numerical results of unsteady
flows in the channel are presented for inlet Mach number M∞ 0.012, Reynolds number
Re∞ 4481, and the wall motion frequency 100 Hz.
1. Introduction
A current challenging question is a mathematical and physical description of the mechanism
for transforming the airflow energy in human vocal tract convergent channel into the
acoustic energy representing the voice source in humans. The voice source signal travels from
the glottis to the mouth, exciting the acoustic supraglottal spaces, and becomes modified by
acoustic resonance properties of the vocal tract 1. The airflow coming from the lungs causes
self-oscillations of the vocal folds, and the glottis completely closes in normal phonation
regimes, generating acoustic pressure fluctuations. In this study, the movement of the
boundary channel is known, harmonically opening and nearly closing in the narrowest crosssection of the channel, making the investigation of the airflow field in the glottal region
possible.
2
Journal of Applied Mathematics
Acoustic wave propagation in the vocal tract is usually modeled from incompressible
flow models separately using linear acoustic perturbation theory, the wave equation for the
potential flow 2, or the Lighthill approach on sound generated aerodynamically 3.
Goal of this work is numerical simulation of compressible viscous flow in 2D
convergent channel which involves attributes of real flow causing acoustic perturbations as
is “Coandă phenomenon” the tendency of a fluid jet to be attracted to a nearby surface,
vortex convection and diffusion, jet flapping, and so forth along with lower call on computer
time, due to later extension in 3D channel flow. Particular attention is paid to the analysis of
acoustic pressure signal from the channel.
2. Mathematical Model
To describe the unsteady laminar flow of a compressible viscous fluid in a channel, the 2D
system of Navier-Stokes equations was considered as a mathematical model. The system of
Navier-Stokes equations is expressed in nondimensional conservative form 4
1 ∂R ∂S
∂W ∂F ∂G
,
∂t
∂x ∂y
Re ∂x ∂y
2.1
where W ρ, ρu, ρv, eT is the vector of conservative variables, F and G are the vectors of
inviscid fluxes, and R and S are the vectors of viscous fluxes. The static pressure p is expressed
by the state equation in the form
1 p κ − 1 e − ρ u2 v2 .
2
2.2
The transformation to the nondimensional form uses inflow parameters marked with the
infinity subscript as reference variables dimensional variables are marked with the hat:
r 0.02 m,
the speed of sound c∞ 343 ms−1 , density ρ∞ 1.225 kg m−3 , reference length L
−6
and dynamic viscosity η∞ 18 · 10 Pa · s. Inflow temperature T∞ K depends on relation
2
T∞ where κ 1.4 is the ratio of the specific heats. Inflow pressure
κR
for speed of sound c∞
T∞ .
satisfies equation of state for ideal gas p∞ ρ∞ R
General Reynolds number in 2.1 is computed from reference variables Re ρ∞ c∞ Lr /η∞ . The nondimensional dynamic viscosity in the dissipative terms is a function
of temperature in the form η T/T∞ 3/4 . The heat transfer coefficient is expressed as
k ηκ/Prκ − 1, where Pr 0.7 is the Prandtl number.
3. Computational Domain and Boundary Conditions
For phonation of vowels, the frequencies of the vocal folds oscillations are in the region from
cc 82 Hz for bass up to cc 1170 Hz for soprano in singing voice, and the airflow velocity in the
trachea is approximately in the range of 0.3–5.2 ms−1 taking into account the tracheal diameter
in humans in the range 14.5–17.6 mm 2.
The bounded computational domain D1 , used for the numerical solution of flow field
in the channel, is shown in Figure 1. The domain is a symmetric channel, the shape of which is
Journal of Applied Mathematics
3
g
H
D1
C
A
B
L
Figure 1: Computational domain D1 . L 8 160 mm, H 0.8 16 mm, and g 0.08 1.6 mm—middle
position.
inspired by the shape 5 of the trachea inlet part of the channel, vocal folds, false vocal folds
and supraglottal spaces outlet part. The upper and the lower boundaries are the channel
walls. A part of the wall changes its shape between the points A and B according to a given
harmonic function of time and axial coordinate see, e.g., 6. The gap gt is the narrowest
part of the channel in point C. The gap width was oscillating with frequency 100 Hz typical
for normal male voice between the minimum gmin 0.4 mm and maximum gmax 2.8 mm,
not closing the channel completely.
The boundary conditions are considered in the following formulation:
∞ /
c∞ M∞ , v∞ 0, ρ∞ 1, and p∞ is extrapolated
1 upstream conditions: u∞ u
from domain D1 ;
2 downstream conditions: p2 1/κ and ρ, ρu, ρv are extrapolated from D1 ;
3 flow on the wall: u, v uwall , vwall where uwall , vwall is velocity vector of the
wall and ∂T/∂
n 0 where T κp/ρ is the temperature.
The general Reynolds number in 2.1 multiplied with nondimensional value M∞ H
represents kinematic viscosity scale, and for computation of the real problem, inlet Reynolds
r /η∞ is used.
number Re∞ ρ∞ c∞ M∞ H L
4. Numerical Solution
The numerical solution uses FVM in conservative cell-centered form on the grid of
quadrilateral cells, see, for example, 4.
The bounded domain is divided into mutually disjoint subdomains Di,j i.e.,
quadrilateral cells. The system of 2.1 is integrated over the subdomains Di,j using the
Green formula and the mean value theorem. In the time-changing domain, the integral form
of FVM is derived using the ALE formulation. The ALE method defines homeomorphic
mapping of the reference domain Dt0 at initial time t 0 to a domain Dt at t > 0 7.
4
Journal of Applied Mathematics
∆y2
3
k=2
k=1
k=
Di, j
s1
l=
4
y, j
l=
R
k=4
Di+1, j
1
3
s4
2
s3
P
l=
Q
Di−1, j
Dual volume V1
s2
l=
∆x2
Di, j+1
O
Di, j−1
x, i
Figure 2: Finite volume Di,j and dual volume Vk .
The explicit predictor-corrector MacCormack MC scheme in the domain with a
moving grid of quadrilateral cells is used. The scheme is 2nd order accurate in time and
space using orthogonal regular grid 4
Wn1/2
i,j
n1
Wi,j
4 Δt 1 n
1 n
n
n
n
n
S Δxk ,
− n1
Fk − s1k Wk −
R Δyk − Gk − s2k Wk −
Re k
Re k
μi,j k1
4 Δt n1/2 − s1k Wn1/2 − 1 R
n1/2 Δyk
−
F
n1 Wni,j Wn1/2
i,j
k
k
Re k
μi,j 2
2μn1
i,j k1
n1/2 − s2k Wn1/2 − 1 S
n1/2 Δxk ,
− G
k
k
Re k
4.1
μni,j
Wni,j
μn1
i,j
μni,j 1 where Δt tn1 − tn is the time step, μi,j Di,j dx dy is the volume of cell Di,j , Δx and Δy
are the steps of the grid in directions x and y, and vector sk s1 , s2 k represents the speed of
edge k see Figure 2. The physical fluxes F, G, R, S on the edge k of the cell Di,j are replaced
R,
as approximations of the physical fluxes.
G,
S
by numerical fluxes marked with tilde F,
The approximations of the convective terms sWk and the numerical viscous fluxes
k on the edge k are central. The higher partial derivatives of velocity and temperature
k, S
R
k are approximated using dual volumes V see 4 shown in Figure 2. The inviscid
k, S
in R
k
numerical fluxes are approximated by the physical fluxes from the cell on the left side of the
current edge in the predictor step and from the cell on the right side of the current edge in
the corrector step.
The last term used in the MC scheme is the Jameson artificial dissipation ADWi,j n
8. Artificial dissipation is used to stabilize computation and also due to velocity gradients
in the narrowest width of the channel, where M∞ ≈ 0.5. The vector of conservative variables
n1
n
is computed at a new time level tn1 : Wn1
i,j Wi,j ADWi,j .
Journal of Applied Mathematics
5
0.844
0.84
0.842
0.82
0.84
0.8
0.838
0.78
0.836
0.76
0.834
2.25
2.3
2.35
2.295
a
2.3
2.305
2.31
b
Figure 3: Grid of quadrilateral cells in the narrowest part of domain D1 at the middle position of the gap
.
width g 0.08 1.6 mm. Detail: Δymin 5 · 10−4 0.01 mm.
The stability condition of the scheme on the regular orthogonal grid limits the time
step
Δt ≤ CFL
2
|umax | c |vmax | c
Δxmin
Δymin
Re
1
1
2
2
Δxmin Δymin
−1
,
4.2
where c denotes the local speed of sound, umax and vmax are the maximum velocities in the
domain, and CFL < 1 for nonlinear equations 9. Time discretization of the scheme satisfies
discrete geometric conservation law DGCL, see 10.
The grid used in the channel has successive
refinement cells near the wall. The
minimum cell size in y-direction is Δymin ≈ 1/ Re∞ to capture the boundary layer effects.
Figure 3 shows the detail of the grid in domain D1 in the narrowest channel cross-section at
the middle position of the gap.
5. Numerical Results
The numerical results were obtained using a specifically developed program for the
u∞ 4.116 ms−1 , Reynolds number
following input data: Mach number M∞ 0.012 p2 102942 P a at the outlet, and wall oscillation
Re∞ 4481, atmospheric pressure p2 1/κ frequency f 100 Hz. The computational domain contained 450 × 100 cells in D1 .
The computation has been carried out in two stages. First, a steady numerical solution
is obtained, when the channel between points A and B has a rigid wall fixed in the middle
position of the gap width. Then this solution is used as the initial condition for the unsteady
simulation.
Figure 4 shows the initial condition for unsteady computation of the flow field in
domain D1 and the convergence to the steady-state solution computed using the L2 norm
of momentum residuals ρu. The maximum Mach number computed in the flow field
max 60.7 ms−1 .
Figure 4a was Mmax 0.177 corresponding to the dimension velocity u
The picture displays nonsymmetric flow developed behind the narrowest channel crosssection. The graph in Figure 4b indicates the nonstationary solution of initial condition
6
Journal of Applied Mathematics
0.06
0.1
0.14
0.18
0.22
0.26
0.3
0.34
0.38
0.42
0.46
−3.5
0.5
Res
Ma: 0.02
−4.5
1
0
−4
0
0
2
4
6
a The velocity flow field mapped by isolines of Mach number
2E+06
4E+06
6E+06
lt
b Convergence to the steady-state solution using
L2 norm of momentum residuals ρu
Figure 4: The initial condition computed in D1 —M∞ 0.012, Re∞ 4481, p2 1/κ, 450 × 100 cells, and
Mmax 0.177 umax 60.7 ms−1 .
which is caused probably by eddies separated in the unmovable glottal orifice and floating
away.
The numerical simulation of the airflow computed in domain D1 over the fourth cycle
of the wall oscillation is presented in Figure 5 showing the unsteady flow field in five time
instants during one vibration period. Large eddies are developing in supraglottal spaces and
a “Coandă” effect is apparent in the flow field pattern. The absolute maximum of Mach
umax 183.5 ms−1 in the flow field during fourth cycle was achieved
number Mmax 0.535 at time t 34.2 ms g 1.002 mm, opening phase behind the narrowest channel crosssection. The flow becomes practically periodic after the first period of oscillations.
Figure 6 shows three vibration periods of the gap width oscillation a and the acoustic
pressures signals computed in the gap b and at the outlet c on the axis of the channel. The
acoustic pressure was calculated by subtracting the average values of the pressure signals
pac p − p2 . The acoustic pressure time dependent data are transformed to frequencydependent data acoustic pressure spectrum using discrete fourier transformation DFT
of the signals. The spectrum of the pressure in the glottis, Figure 6b-right, shows the
dominant fundamental frequency of vocal folds model oscillations f0 100 Hz and the
generated higher harmonics as a consequence of the throttling and nearly closing the glottal
gap gt. Two different regimes are apparent in the acoustic pressure signal at the channel
outlet during one vibration period of the glottis in Figure 6c-left. Relatively smooth signal
containing low frequencies is dominant in the time interval corresponding to the phase of
maximum glottal opening, and a very noisy signal containing high frequencies is associated
with the phase of minimum glottal opening. Four acoustic resonances of the channel cavity
at about f0 100, f1 550, f2 1150, and f3 1950 Hz can be identified in the spectrum
envelope of the pressure at the channel outlet in Figure 6c-right. The first acoustic resonance
f1 550 Hz see spectrum peak at cc 500 Hz corresponds to the first eigenfrequency F1
of a simple tube of the length of the complete channel closed at the inlet and open at the
r 343 ms−1 /4 · 0.16 m 536 Hz. The acoustic resonances are
outlet F1 c∞ /4L · L
more damped with increasing frequency, which can be caused by the fluid viscosity as well
as by a numerical viscosity implemented in the numerical method constants magnitude in
ADWi,j n .
Remark 5.1. We used several tests on fine and coarse grids in the computational domain. Also
the domain has been prolonged in upstream and also in downstream part. Achieved results
were approximately the same on fine grid.
Journal of Applied Mathematics
7
1.6
1.2
0.8
0.4
0
0
2
4
6
8
a t 30 ms, g 1.6 mm, Mmax 0.159 54.5 ms−1 1.6
1.2
0.8
0.4
0
0
2
4
6
8
−1
b t 32.5 ms, g 0.4 mm, Mmax 0.236 80.9 ms 1.6
1.2
0.8
0.4
0
0
2
4
6
8
−1
c t 35 ms, g 1.6 mm, Mmax 0.370 126.9 ms 1.6
1.2
0.8
0.4
0
0
2
4
6
8
d t 37.5 ms, g 2.8 mm, Mmax 0.097 33.3 ms−1 1.6
1.2
0.8
0.4
0
0
2
4
6
8
−1
e t 40 ms, g 1.6 mm, Mmax 0.162 55.6 ms Figure 5: The unsteady numerical solution of the airflow in D1 — f 100 Hz, M∞ 0.012, Re∞ 4481,
p2 1/κ, and 450 × 100 cells. Data computed during the fourth oscillation cycle. Results are mapped by
isolines of velocity ratio and by streamlines.
Remark 5.2. The mathematical model 2.1 of laminar flow used in this case is debatable. For
the first approximation, we supposed unformed turbulent flow at the inlet part of the channel.
Remark 5.3. The validation of computations for this case is not complete because of
experiments absence. Semivalidation of the computations is comparison with particle image
velocimetry method PIV experiment, but we can compare only qualitative behavior of the
flow. Full validation of the code for subsonic and transonic flow through a turbine cascade
computed in periodic domain is showed, for example, in 10.
Journal of Applied Mathematics
Gap (mm)
8
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0
0.005
0.01
0.015
0.02
0.025
0.03
(s)
a The prescribed oscillation of gap width f 100 Hz.
10000
5000
8000
4000
3000
(Pa)
(Pa)
6000
4000
2000
1000
0
−2000
2000
0
0.005
0.01
0.015
(s)
0.02
0.025
0
0.03
0
1000
2000
3000
4000
5000
3000
4000
5000
(Hz)
b The acoustic pressure in the gap and DFT spectrum—f0 100 Hz.
30
2.5
20
2
(Pa)
(Pa)
10
0
−10
1
0.5
−20
−30
1.5
0
0.005
0.01
0.015
0.02
0.025
(s)
0.03
0
0
1000
2000
(Hz)
c The acoustic pressure at the outlet and DFT spectrum—f0 100, f1 550, f2 1150, f3 1950 Hz.
Figure 6: Three vibration periods of the gap width oscillations and the acoustic pressure signals pac
computed in the gap and at the outlet on the axis of the channel.
6. Discussion and Conclusions
The numerical solution in the channel showed large vortex structures developed in the
supraglottal space moving slowly downstream and decaying gradually. It was possible to
detect a “Coandă phenomenon” in the computed flow field patterns. A similar generation of
large-scale vortices, vortex convection and diffusion, jet flapping, and general flow patterns
Journal of Applied Mathematics
9
were experimentally obtained in physical models of the vocal folds by using PIV method in
5, 11, 12.
The results show that some numerical results of viscous flow in a symmetric channel
using a symmetric grid and scheme can be nonsymmetrical, depending on the geometry and
the Reynolds number. This effect was observed also for laminar transonic flow computation
13. The assumption of the axisymmetry solution for the axisymmetry channels see 6
excludes modeling the “Coandă” effect and large vortex structures of the size comparable
with the cross-section of the channel.
The analysis of the computed pressure revealed basic acoustic characteristics of the
channel. This is promising result for future studies for a direct modeling of human voice
generated by the flow in vibrating glottis taken into account a real shape of the human vocal
tract for phonation and throttling the glottal gap width up to zero.
Acknowledgment
This paper was partially supported by Research Plans MSM 6840770010, GAČR P101/
11/0207 and 201/08/0012.
References
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City, IA, Iowa, USA, 2006.
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